2012-10 Blog Entry List
日時 １１月１日木曜 １５：１５－１６：３０
ColorSymmetries Associated with Non-Periodic Structures
Ma. Louise Antonette N. De Las Peñas, PhD
Professor, Mathematics Department
Ateneo De Manila University Philippines
With the discovery of quasicrystals in 1984, the research field ofnon-periodic crystallography has grown and expanded in several directions.Structural problems continue to interest mathematicians and physicists.
In this talk, we discuss a method that allows the investigation of symmetriesof non-periodic structures via colorings of cyclotomic integers. In particular,our work looks at ideal colorings of Mn= Z[xn] where xn = e2pi/nis a primitive nth root ofunity for values of n for which Z[xn] is aprincipalideal domain and thus has class number one. The values of n are groupedinto classes with equal value of f(n),the Euler’s totient function. In the lecture, some results on color groups andcolor preserving groups will be presented.
The colorings of Mn may be manifested geometricallyas a vertex or tile coloring of a two dimensional tiling with n-foldrotational symmetry, which is non-periodic for f(n) > 2. For suchcases, since Mn is dense on the plane, we choose a discretesubset of Mn for which we show the colors. The discovery ofquasiperiodic tilings such as the Penrose tiling, also raised the questionabout color symmetries of such tilings.